By Theodore W. Gray and Jerry Glynn, published by Cambridge University
Press

In an otherwise business-like introduction to
Mathematica
, Jerry and I decided to include a couple of pretty hot-headed chapters about
education and software. Here they are, minus the level-headed introduction
to Mathematica.

Will it rot my students' brains if they use Mathematica?

Jerry: I have young students who reach for their calculators
to get the answer to 5×6.

Theo: I didn't say incompetence is a sufficient condition
for progress. I do say it is a necessary one. If
you want to move beyond endless drudgery, you have to have technology (or
slaves, servants, or a spouse) to free you from the otherwise all-consuming
task of survival. Technology is the least-objectionable way of
generating free time, in my opinion. Of course, some people will
use their free time more responsibly than others.

Jerry: People are very attached to the value of their
skills. They believe that the skills of their generation should
be preserved, with new skills added on.

Theo: Such an attitude represents a tremendous degree of disrespect
of our forepersons. It was really, really hard to be a
cave person. The skills needed to live comfortably in, say, northern
Europe in 20,000 BCE were extremely complex. They required then
and would require now the full range of human intelligence.

To think that a modern human should be able to do everything that previous
generations have been able to do (hunt, speak Latin, do square roots by hand,
etc.), and also have any time left over to learn anything new (microbiology,
email, calculus), is basically insulting to all those previous generations,
since it implies that they under-employed their intelligence. It
is also quite false.

Jerry: I think it matters that students spend their time thinking
and learning. People seem happiest if they are good at something. But
I agree it doesn't matter whether they learn all the same things their parents
learned. Not learning Latin is a problem only if you need to speak
to Latin people on a regular basis, or if people will make fun of you on
the playground. Not learning to add is a problem only if you have
to add regularly, or if people will make fun of you for using a calculator
to do

Theo: Well, you probably do think people should learn
to add. Adding is not that hard, and it's a fairly practical skill
in the day-to-day world.

Jerry: In the old days (before television), being able to add
up a long column of numbers without making any mistakes was a valuable skill. People
would pay you a living wage to do nothing but add numbers well. Not
today.

Theo: Today, it's nice to be able to add small numbers, and larger
numbers in a pinch, but the specific mental tricks and habits needed to get
the right answer consistently when adding lots of numbers are just not helpful. Not
being able to do this does not represent a failure of the intellect, any
more than not knowing which fields in your neighborhood have the best rabbit
hunting: both were, at one time, failings that would get you laughed at.

Theo:No, and let me make this very clear. No
one can learn to think without having something to think about. If
you try to teach someone how to think in the abstract, you are not
going to get anywhere. If you try to make education "easy", by
removing the content, you are cheating your students out of the most important
thing you have to offer: the chance to do something hard. Only
by mastering a difficult body of knowledge can a child develop into a confident,
thinking adult. The point is, it doesn't necessarily have to be
the same difficult body of knowledge that the child's parents learned.

If you start a lesson off by telling the students "This is going to
be easy", you are simultaneously telling them "We had to make this easy because
we don't think you're capable of doing anything hard". And when
the lesson is over, the only sense of accomplishment they can feel is that
they did something easy. So what?

Learning is hard work. If you are not working hard, you are
not learning. Period. Kids love hard work, as long
as they see where it's going and why. Instead of killing that
energy by giving them something easy, we should foster it by giving
them something really hard. We should tell them it's hard. We
should give them the chance to do something meaningful.

Jerry: Readers should be aware that Theo is the father of one
three year old and a couple-odd babies, while I am the father of four adults. It
is well known that people at the beginning of the child rearing process have
much stronger opinions than those who have completed at least two children. However,
in this case I have to agree with Theo.

His viewpoint is nicely supported by Joseph Mitchell's story in his wonderful
book Up in the Old Hotel about a bridge-building disaster which killed
a number of young members of the Mohawk nation. People believed
this disaster would drive the Mohawks away, but instead it had the opposite
effect. The tragedy confirmed that working on high steel is serious, dangerous
work, worthy of the efforts of young Mohawk men. Mohawks have
gone on to erect the steel of a high proportion of all the high-rise buildings
and bridges on the North American continent since 1910.

We seem to agree in general principles but how do we put them into action. If
we want our children to learn some things that are hard, how do we decide
which hard things are-- or are not--worth their effort?

Theo: It's not easy, and of course it changes every day. The
one sure wrong answer is: The same thing we learned as children.

Therapy can probably help, if you feel you must insist that your child
learn the multiplication tables up to 12 by 12 because you had to,
and you would be embarrassed if her grandma finds out. If you
want her to learn it because you really think it's something that will help
her in her life, fine, there's nothing wrong with that. Examine
your motives. Look around at our society. Be open-minded.

Jerry: I forget, what was the point of this chapter? Have
we said anything yet?

Theo: The point of this chapter is to tell people that if they
think Mathematica might be harmful to the process of education, they
need to reconsider their fundamental belief structure. Just like
the breech-loading rifle, or the pocket calculator, modern tools such as
Mathematica change (maybe a little, maybe a lot) the kinds of things
that ought to be learned. Some things that used to be important
are not anymore, and some new things have become important.

Jerry: How about a short list? After Mathematica
:

Less Important
Guessing factors for polynomials
Knowing many tricks for integration
Being careful when copying over expressions many times
Finding roots of complex equations
Knowing how to do matrix row operations
Knowing how to avoid dropping minus signs
Memorizing specific rules for derivatives of such functions as tangent and
secant
Memorizing multiple angle formulas for trig functions

More Important
Translating statements about problems in natural language into statements
in mathematical or procedural language
Learning how to experiment with math
Knowing which integrals should best be done numerically
Knowing how to work backwards or to use numerical methods to check symbolic
results for plausibility
Knowing how to use techniques from programming
Understanding recursion and how to use it practically
Knowing which functions are discontinuous and where they are discontinuous
Knowing how to mix math and programming

Theo: So, won't people be in trouble if they don't know how to
do these traditional, now less important, things? What if they
have to function without computers some day?

Jerry: Readers who think the topic of this chapter is unimportant
are mistaken. Computer use in our schools today can only be described
as pathetically limited. Attitudes we have discussed here are
a serious factor. Maybe ten years ago one could argue that computers
did not belong in the classroom. They were too expensive, good
software was not available, and they were too difficult to operate. But
today, a math classroom without a computer is a joke. Yet the
number of math teachers who use computers regularly with their students is
very small. In same cases, teachers have computers in the
classroom, but by the time they are finished teaching all the old lessons,
there is no time left over for using the computers. Try to tell
them to cut out some of the old topics, and war breaks out. In
that conflict, parents and teachers are often on the same side, outnumbering
the voices for change. Misuse of computers is easy and a problem but no
use is presently a bigger problem.

Will it rot my students' brains if they use other educational software?

Theo: Yes.

Jerry: But wait, didn't we just write a whole chapter about how
great it was for students to forget everything their parents ever learned
and use Mathematica instead?

Theo: Well, yes, but we were talking about one particular piece
of software being used by relatively advanced students to replace manual
computation, a very narrow set of skills.

It is interesting to note that many teachers and parents feel that using
calculators to avoid learning multiplication tables, or using Mathematica
to avoid learning integration, is an educational failure. Yet
they don't seem to have a problem with using software that retards development
of far more fundamental aspects of the student's humanity, such as creativity,
enthusiasm, love of learning, and empathy.

The sad fact is that 95% of all "educational software" and 95% of all
educational use of computers is, for lack of a better word, crap. I'm
not claiming that Mathematica is the only good educational software
in the world. But you have to look far and wide to find the few
others.

Jerry: Why do you say it's mostly crap? And if you're
right, how did educational software get to be so bad?

Theo: It's bad because thousands of earnest and dedicated software
engineers are working hard to make it that way. Bringing to bear
all their skills in commercial software design, the best programmers in the
industry have created an impressive body of absolutely worthless educational
software. Worse than worthless, much of it is downright scary
in its effects.

Jerry: How about some specific examples. So far this
is sounding like a pretty categorical attack on the competition by someone
with a vested interest in one particular product.

Theo: Video games are at one end of a continuum. Educational
software doesn't usually have the same kind of violence, and it tries to
be "positive". But the fact is that it's written by the same people,
and more importantly, it's often judged by the same criteria, as video games
and other types of entertainment software.

When software engineers design educational software, there are a few
things they take for granted, and reviewers of educational software appear
to agree with them:

Software should be "easy to use". That means you shouldn't
have to read a manual before starting to use it; you should be able to click
on just about anything on the screen and have it work; you should be able
to figure out how to use the program just by flipping through the menus and
trying things that seem relevant. In short, you should be able
to use the software by trial and error.

Users should never feel at a loss for what to do next. The
software should lead them seamlessly from one stage to the next, with no
moments of confusion or uncertainty.

Software should engage and keep its users' attention with colorful graphics,
sounds, animations, whatever it takes. Software without these
elements is judged to be boring and of inferior quality to the better grades
of multi-media educational software.

Good performance should be rewarded with a treat, such as a clever animation
or a game that can be played for a while.

Obvious, right? Any software that lacks these characteristics
is routinely trashed in reviews. It is called "hard to use", "boring",
or "unmotivating". We are told repeatedly that kids prefer flashy
graphics software.

Well, kids prefer TV over homework, late movies instead of sleep, and
chocolate coated sugar bombs for breakfast. Have we all forgotten
what education is supposed to be about? Shouldn't education be
about preparing children to lead successful, fulfilling lives? Here
are a few self-evident facts about children and learning:

Children need to know certain facts, but acquiring them is not the main
point of learning, especially not in the earlier grades. Mainly
children need to have effective habits of mind and an ability to think analytically. They
also need to be self-motivated, because in real life there isn't always someone
there to provide external motivation (unless they join the army).

The hardest things to teach are the skill of solving problems with incomplete
information, the skill of figuring out which problems need solving in the
first place, and the skill of finding and bringing together the resources
needed to solve a problem.

Children are primed to want to learn. They start out valuing
learning and accomplishment above anything else in the world. If
you see a child uninterested in learning, it is overwhelmingly likely that
the child was made that way by something in the child's world: Children do
not start out that way. (Of course there are always exceptions,
but they are just that: exceptions.)

Software should be "easy to use".
Life is not "easy to use". Children quickly learn that the best
and most effective way to handle typical educational software is to click
as fast as you can on anything you can find on the screen until something
works. It's really not worthwhile to read the bothersome text;
understanding the material just slows you down. Since the software
is "high quality", it will always do something, help you out, not be judgmental;
pretty soon you'll hit the right icon and get to the next level.

Extensive use of this sort of software supports and develops shallow
thinking habits. It discourages careful analytic thought. It
is dysfunctional. Real life is not a multiple choice test. In
real life there are points off for making wrong choices.

The user should never feel at a loss for what to do next.
In life, it's rarely clear what to do next. Uncertainty
is one of the defining characteristics of life; one of the most important
things a good education can give a child is the ability to sift through seemly
limitless options and make wise choices. It's usually not even
clear when a choice is there to be made, let alone what the alternatives
are.

Now, it's possible that children have always been helpless and confused
in the real world, but this sort of software isn't helping things. Sure,
some kids will come out fine, and yes, some kids would never have come out
fine. But what about the ones in between who would have developed
into thoughtful, capable adults but for their exposure to this sort of software? How
many is too many?

Software should engage and keep the user's attention.
Good performance should be rewarded with a treat.
These two are without a doubt the most damaging attributes of most educational
software. It is absolutely tragic that, by rewarding learning
with stupid tricks and games, such software devalues the learning itself. Children
who would gladly learn for the sake of learning are, quickly and devestatingly,
turned into children who demand rewards for learning.

Jerry: Well, I can see you feel strongly about this. It
must be nice to be so sure of things.

I think there's quite a bit of software out there that is valid and
valuable. Perhaps we can help people see the difference. I
assume that your attack on "easy to use" software doesn't mean you think
good software should be intentionally hard to use?

Theo: I think that learning is intrinsically hard. Educational
software that students just get through without working hard has not succeeded
in making learning easy, it has simply replaced learning with entertainment.

Imagine taking your kids to a fun movie, having a good time, eating
ice cream. No harm in that. But should you tell them
that they are having an important educational experience? Should
you tell them that they have accomplished something by sitting through
the whole movie? What a horrible devaluation of accomplishment
that would be! What's the point of actually working hard to reach
success, if your parents and teachers will praise you for joking around and
eating ice cream?

That's exactly what most educational software is: A stupid, insipid
movie that keeps your kids entertained for an hour or two while the teacher
plans a bake sale to replace the music budget that was spent on computers. Kids
don't learn anything, but worse they are being taught that it's not necessary
to work hard to achieve success.

Software should not be unnecessarily hard to use, but neither
should it shy away from or disguise the inherent richness of the subject
matter. It should be open-ended, deep, and capable of doing senseless
things if asked.

Jerry: Why should software be able to do senseless things? What's
the point in that?

Theo: Let's take a specific case. Give me an example
of a program you like.

Theo: Exactly! If students decide to build a completely
useless geometrical construction, the program won't stop them. It
lets them discover for themselves that their construction is uninteresting. This
is very important: By allowing freedom to go off in the wrong direction,
the software is giving students the opportunity to learn.

Jerry: Maybe I'm starting to see what you mean. As
the saying goes, you learn from your mistakes. Software that prevents
you from making mistakes limits what you can learn.

Theo: Exactly. I'm not saying that educational software
should encourage mistakes, or have outright flaws, such as buttons that are
confusingly labeled and crash the program. But I am saying that
programs should be rich enough to allow both right and wrong paths to be
followed, and followed in a more than superficial way.

Simulation software in contrast has a set of rules and algorithms (the
laws of physics; sociological models; geometrical relationships; etc.). It
is able to apply these rules to a fairly open-ended set of inputs. Simulation
software is inherently more difficult to write; not surprisingly there is
little of it out there. In fact, good simulations are so difficult
to create that one could almost name all the examples that have ever been
done. Page-turning software is incredibly easy to make, and there
are countless thousands of titles available, virtually all of very low quality.

Interestingly, while violent video games may be evil, they are largely
simulation-based. No page-turning software could hold anyone's
attention for the hundreds of hours that video games capture our children. In
a good video game, you have a huge world to roam about in, complete with
back alleys, multiple levels, and great detail in all the parts. The
only problem is that you have to keep killing people to see the next back
alley. Oh well, if people get in your way, killing them is OK,
right?

Jerry: OK, OK, let's not get started on that again. Now,
by that definition, Mathematica is simulation software, as is most
of the educational software I like. But this kind of software
is much harder on the teachers. Students are constantly going
off on tangents and asking questions about what they've done.

At some schools, teachers (and maybe administrators) would rather use
software that the students can use for an hour without needing a teacher. From
their point of view, what's the point of spending your school's entire music,
PE, and after-school programs budget on a shiny new computer lab, if you
just have to hire new teachers to baby-sit the students? For that
kind of money, shouldn't it be "automatic"?

Finding and training teachers able to work with students using untamed,
open-ended software is very hard. Potential teachers may be computer-phobic,
or may not want to be put in the position of answering questions instead
of lecturing. It's much harder to answer students' open-ended
questions than it is to spill out a prepared lecture. They may
not always know the answers to all the questions; from time to time, a student
might actually end up knowing more than the teacher does about something.

Theo: I think we can all agree that it would be sad to see classroom
teaching replaced by computer lab brain damage, just because school administrations
don't know what they are doing. It would be far better not to
bother with the computers in the first place, if they're not going to be
used in ways that help people learn.